Skip to main content

Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

Print preview

Paper size
Additional printing options
Print headers
Print footers

Section BF-3 Difference Quotients and Piecewise Functions

Worksheet BF-3 Quick Notes

For this outcome we will be looking at relations and functions. We will evaluate functions (specifically looking at average rate of change and the difference quotient). We will also look at graphs of basic functions and discuss properties of functions. We will look at how these can be broken up and combined to form piecewise functions. Specifically, for this outcome, students should be able to:
  • Use and understand function notation.
    🔗
    🔗
  • Find and interpret a function’s average rate of change over an interval.
    🔗
    🔗
  • Set up an evaluate various types of difference quotients.
    🔗
    🔗
  • Given a piecewise function:
    • State the simplified domain in interval notation.
      🔗
      🔗
    • Find the intercepts.
      🔗
      🔗
    • Find and label the boundary points.
      🔗
      🔗
    • Graph the function with correct notation on the boundary points, correct slope, and general behavior of the graph.
      🔗
      🔗
    🔗
    🔗
🔗
Functions
🔗
A function is a set of ordered pairs that have the property that no input value has more than one corresponding output value. We denote a function using the name of the function, the domain variable, and the expression. \(f(x)=2x+3\) is an example of a function named \(f\text{,}\) with domain variable \(x\text{,}\) defined by the expression \(2x+3.\)
🔗
To evaluate a function we can replace the domain variable with the number or expression to be evaluated.
🔗
Difference Quotients
🔗
The difference quotient is another way to represent an average rate of change. Typically this average rate will be measured over an interval, so given an interval \([a,b]\) and a function \(f(x)\) we say that the average rate of change of \(f\) on \([a,b]\) is given by \(ARC=\frac{f(b)-f(a)}{b-a}\text{.}\) If we don’t have the full interval but know the function and the change, \(h\text{,}\) we can write this as \(ARC=\frac{f(x+h)-f(x)}{h}\text{.}\)
🔗
The difference quotient can be written in various forms but will always be a change in the function value divided by the change in the domain values.
🔗
Piecewise Functions
🔗
A piecewise function (or piecewise defined function) is a function defined by multiple subfunctions, each applying to a subset of the overall domain.
🔗
How to Graph a Piecewise Function
  1. Identify Intervals: Determine the \(x\)-intervals for each piece.
    🔗
    🔗
  2. Graph Each Piece: Sketch each individual formula (e.g., lines, parabolas) only within its specified interval.
    🔗
    🔗
  3. Check Boundaries: Use a filled circle \(\bullet\) for \(\le\) or \(\ge\) (included) and an open circle \(\circ\) for \(\lt\) or \(\gt\) (excluded).
    🔗
    🔗
  4. Verify Continuity: Note if the graph has breaks (discontinuities) or connects between intervals.
    🔗
    🔗
🔗
🔗

Section BF-3 Videos

Slopes and Average Rates
🔗
Average Rates of Change
🔗
Basic Graphs of Functions
🔗
Piecewise Functions
🔗
🔗

Section BF-3 Rubric

Worksheet Worksheet

Before attempting the BF-3 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
🔗
Table 12. BF3 Rubric
🔗
Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner No algebraic errors OR some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Basics of Graphing Graph is labeled with units for each axis. The graph includes all of the important features (intercepts, asymptotes, deleted points, etc).  The graph is appropriately drawn to accommodate all features. Graph is labeled with units for each axis and includes most of the important features or the graph has all of the important features but is not labeled. Graph is not labeled and has some of the important features. No graph was given or the given graph is not labeled and does not have the correct features of the function.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution. Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution. Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
Difference Quotient Correctly set up and simplified the requested difference quotient. Used correct notation for the function and the answer. Correctly set up the requested form of the difference quotient, but had some errors in simplifying. Used correct notation for the function and the answer. Set up a difference quotient, but did not fully evaluate or made errors in evaluating the functions. Some errors in simplifying. Did not label the function or had errors in the function notation. No progress beyond rewriting the problem. Not enough evidence to show knowledge of how to set up or simplify a difference quotient.
Piecewise Functions Provided correct, simplified domain using correct interval notation. Both x and y intercepts were given in correct notation. All of the boundary points were stated and correctly identified on the graph. Graph contains all relevant information; has correct end-behavior, correct slopes, and correct boundaries. If there is symmetry, the correct symmetry is presented. Provided correct domain using interval notation. Either the x or y intercepts were given with correct notation. Some of the boundary points were given and correctly identified on the graph OR all of the boundary points were given but not correctly identified on the graph. Graph contains most of the relevant information; it has partially correct end-behavior, slopes, and boundaries. If there is symmetry, the correct symmetry is presented. Provided a domain with errors or the domain was not given in interval notation. Either intercepts were not given, points that are not intercepts were given, or both types of intercepts were incorrect. Boundary points were missing or incorrectly given and incorrectly identified on the graph. Relevant parts of the graph are missing. errors in the end-behavior, slopes, and boundaries. No domain, intercepts, or boundary points were given.  No graph was given.
🔗
🔗

Worksheet BF-3 Sample Outcomes

This outcome covers difference quotients and piecewise functions. Given a function \(f(x)\text{,}\) you should be able to compute a difference quotient, such as: \(\frac{f(x+h)-f(x)}{h}\) and simplify it. Given a piecewise function, you should be able to find the domain, the boundary points, the intercepts, and sketch the graph. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.
🔗
For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
🔗

1.

Given the function \(f(x)=\frac{x}{x-3}\text{.}\) Find and simplify the difference quotient \(\frac{f(x+h)-f(x)}{h}\text{.}\)
🔗
Hint.
Start by evaluating both \(f(x+h)\) and \(f(x)\) separately. Then simplify the resulting expression.
🔗
🔗
Answer.
\(\frac{f(x+h)-f(x)}{h}=\frac{3}{(x+h-3)(x-3)}\)
🔗
🔗
Solution.
To begin we will start by evaluating each function: \(f(x+h)=\frac{x+h}{x+h-3}\) and \(f(x)=\frac{x}{x-3}\text{.}\) Putting these together and simplify the expression.
\begin{align*} \begin{aligned}\frac{f(x+h)-f(x)}{h} =& \frac{\frac{x+h}{x+h-3}-\frac{x}{x-3}}{h}\\ =& \frac{\frac{(x+h)(x-3)-x(x+h-3)}{(x+h-3)(x-3)}}{h}\\ =& \frac{\frac{3h}{(x+h-3)(x-3)}}{h}\\ =& \frac{3}{(x+h-3)(x-3)}\end{aligned} \end{align*}
🔗
🔗
🔗

2.

Given the function \(f(x)=4x^2-5x\text{.}\) Find and simplify the difference quotient \(\frac{f(a)-f(3)}{a-3}\text{.}\)
🔗
Hint.
Start by evaluating both \(f(a)\) and \(f(3)\) separately. Then simplify the resulting expression.
🔗
🔗
Answer.
\(\frac{f(a)-f(3)}{a-3}=4a+7\)
🔗
🔗
Solution.
To begin we will start by evaluating each function: \(f(a)=4a^2-5a\) and \(f(3)=4(3)^2-5(3)=36-15=21\text{.}\) Putting these together and simplify the expression.
\begin{align*} \begin{aligned}\frac{f(a)-f(3)}{a-3} =&\frac{4a^2-5a-21}{a-3}\\ =& \frac{(4a+7)(a-3)}{a-3}\\ =& 4a+7\end{aligned} \end{align*}
🔗
🔗
🔗

3.

Graph the function
\begin{equation*} f(x)=\begin{cases}5-3x \text{ }& \text{ }x\le -1\\x^2+1 \text{ }& \text{ } -1 \lt x\lt 2\\ 3x+2 \text{ }& \text{ } x\ge 2\end{cases} \end{equation*}
Also state the domain, boundary points and intercepts.
🔗
Hint.
Start by finding the boundary points and the domain, then graph and find the intercepts.
🔗
🔗
Answer.
Domain: \((-\infty, \infty)\text{,}\) Boundary points: \((-1, 8), (-1,2), (2,5)\) and \((2, 8)\text{,}\) Intercepts: \((0, 1)\)
🔗
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
🔗
Solution.
To graph the function, we will consider each piece separately:
  • For \(x \le -1\text{,}\) the function is \(f(x) = 5 - 3x\text{.}\) This is a linear function with a slope of \(-3\) and a point at \((-1,8)\text{.}\)
    🔗
  • For \(-1 \lt x \lt 2\text{,}\) the function is \(f(x) = x^2 + 1\text{.}\) This is a parabola opening upwards with a vertex at \((0, 1)\) and points at \((-1, 2)\) and \((2, 5)\text{.}\)
    🔗
  • For \(x \ge 2\text{,}\) the function is \(f(x) = 3x + 2\text{.}\) This is a linear function with a slope of \(3\) and a point at \((2, 8)\text{.}\)
    🔗
🔗
🔗
🔗
🔗
🔗
PrevTopNext
Feedback